Common Core · High school
HSF-TF.C.8
High school — Prove the Pythagorean identity sin²θ + cos²θ = 1 and use it to find sine, cosine or tangent given one of them and the quadrant.
The 3 skills that cover it
- Trigonometric IdentitiesUse the Pythagorean and quotient identities to simplify expressions and verify equations.Read the lesson →
- Trigonometric EquationsFind every angle that makes a trigonometric equation true: isolate the ratio, read the reference angle off the unit circle, place it in the right quadrants, and use an identity or a substitution when the equation is quadratic.Read the lesson →
- Half-Angle & Product IdentitiesFinish the identity toolkit: the half-angle formulas and their signs, the power-reducing formulas, product-to-sum and sum-to-product, tan(t/2), and verifying an identity with all of them.Read the lesson →
What a student should be able to do
- Apply sin2 t + cos2 t = 1 and its rearrangements.
- Use the quotient and reciprocal identities.
- Simplify a trigonometric expression to a single function.
- Verify an identity step by step.
- Solve a basic trigonometric equation on a given interval.
- Solve an equation that needs factoring or a quadratic substitution.
- Use an identity to rewrite an equation in one ratio before solving.
- Solve an equation with a multiple angle, and write the general solution.
- Find exact values such as sin 15° and cos(π/8) with a half-angle formula, choosing the sign from the quadrant.
- Given one ratio of t and an interval for t, find sin(t/2), cos(t/2) or tan(t/2).
- Rewrite powers of sine and cosine with the power-reducing formulas, and convert between products and sums.
- Verify an identity or evaluate an expression using the half-angle, product-to-sum and sum-to-product formulas.